Critical window for approximate counting in dense Ising models

2026-03-22

AI Summary

This work investigates the computational complexity of approximately computing the partition function of dense Ising models in the critical regime. Addressing the failure of conventional reduction techniques at criticality, the authors introduce a global fluctuation aggregation method that collectively accounts for statistical fluctuations across all gadgets, rather than imposing concentration constraints individually. This approach yields the first sharp scaling window for the hardness of approximate counting in the critical region. Specifically, they establish that approximating the partition function remains computationally intractable within a critical window of width \(N^{-1/2+\varepsilon}\). The resulting hardness bound nearly matches the best-known algorithmic upper bounds, thereby delineating the optimal computational complexity threshold for this problem at the phase transition critical point.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Computational social science

Abstract

We study the complexity of approximating the partition function of dense Ising models in the critical regime. Recent work of Chen, Chen, Yin, and Zhang (FOCS 2025) established fast mixing at criticality, and even beyond criticality in a window of width $N^{-1/2}$. We complement these algorithmic results by proving nearly tight hardness bounds, thus yielding the first instance of a sharp scaling window for the computational complexity of approximate counting. Specifically, for the dense Ising model we show that approximating the partition function is computationally hard within a window of width $N^{-1/2+\varepsilon}$ for any constant $\varepsilon>0$. Standard hardness reductions for non-critical regimes break down at criticality due to bigger fluctuations in the underlying gadgets, leading to suboptimal bounds. We overcome this barrier via a global approach which aggregates fluctuations across all gadgets rather than requiring tight concentration guarantees for each individually. This new approach yields the optimal exponent for the critical window.

Problem

• approximate counting
• dense Ising models
• critical window
• partition function
• computational hardness